Description: Rearrange arguments in a commutative, associative operation. (Contributed by NM, 26-Aug-1995)
Ref | Expression | ||
---|---|---|---|
Hypotheses | caov.1 | ⊢ 𝐴 ∈ V | |
caov.2 | ⊢ 𝐵 ∈ V | ||
caov.3 | ⊢ 𝐶 ∈ V | ||
caov.com | ⊢ ( 𝑥 𝐹 𝑦 ) = ( 𝑦 𝐹 𝑥 ) | ||
caov.ass | ⊢ ( ( 𝑥 𝐹 𝑦 ) 𝐹 𝑧 ) = ( 𝑥 𝐹 ( 𝑦 𝐹 𝑧 ) ) | ||
Assertion | caov32 | ⊢ ( ( 𝐴 𝐹 𝐵 ) 𝐹 𝐶 ) = ( ( 𝐴 𝐹 𝐶 ) 𝐹 𝐵 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | caov.1 | ⊢ 𝐴 ∈ V | |
2 | caov.2 | ⊢ 𝐵 ∈ V | |
3 | caov.3 | ⊢ 𝐶 ∈ V | |
4 | caov.com | ⊢ ( 𝑥 𝐹 𝑦 ) = ( 𝑦 𝐹 𝑥 ) | |
5 | caov.ass | ⊢ ( ( 𝑥 𝐹 𝑦 ) 𝐹 𝑧 ) = ( 𝑥 𝐹 ( 𝑦 𝐹 𝑧 ) ) | |
6 | 2 3 4 | caovcom | ⊢ ( 𝐵 𝐹 𝐶 ) = ( 𝐶 𝐹 𝐵 ) |
7 | 6 | oveq2i | ⊢ ( 𝐴 𝐹 ( 𝐵 𝐹 𝐶 ) ) = ( 𝐴 𝐹 ( 𝐶 𝐹 𝐵 ) ) |
8 | 1 2 3 5 | caovass | ⊢ ( ( 𝐴 𝐹 𝐵 ) 𝐹 𝐶 ) = ( 𝐴 𝐹 ( 𝐵 𝐹 𝐶 ) ) |
9 | 1 3 2 5 | caovass | ⊢ ( ( 𝐴 𝐹 𝐶 ) 𝐹 𝐵 ) = ( 𝐴 𝐹 ( 𝐶 𝐹 𝐵 ) ) |
10 | 7 8 9 | 3eqtr4i | ⊢ ( ( 𝐴 𝐹 𝐵 ) 𝐹 𝐶 ) = ( ( 𝐴 𝐹 𝐶 ) 𝐹 𝐵 ) |